Abstract A fundamental question in computational nonlinear partial differential equationsis raised to discover if one could construct a functional iterative algorithm for the regularizedlong-wave (RLW) equation (or the Benjamin–Bona–Mahony equation) based on an integralequation formalism? Here, the RLW equation is a third-order nonlinear partial differentialequation, describing physically nonlinear dispersivewaves in shallowwater. For the question,the concept of pseudo-parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul43:118–138, 2017), is introduced and incorporated into the RLW equation. Thereby, dualnonlinear integral equations of second kind involving the parameter are formulated. Theapplication of the fixed point theorem to the integral equations results in a new (semi-analyticand derivative-free) functional iteration algorithm (as required). The new algorithm allowsthe exploration of new regimes of pseudo-parameters, so that it can be valid for a muchwiderrange (in the complex plane) of pseudo-parameter values than that of Jang (2017). Being fairlysimple (or straightforward), the iteration algorithm is found to be not only stable but accurate.Specifically, a numerical experiment on a solitary wave is performed on the convergenceand accuracy of the iteration for various complex values of the pseudo-parameters, furtherproviding the regions of convergence subject to some constraints in the complex plane.Moreover, the algorithm yields a particularly relevant physical investigation of the nonlinearbehavior near the front of a slowly varying wave train, in which, indeed, interesting nonlinearwave features are demonstrated. As a consequence, the preceding question may be answered.
Keywords Regularized long-wave (RLW) equation · Pseudo-parameter ·Nonlinear integralequations of second kind · Functional iteration algorithm
As a starting point of this paper, a fundamental (as well as theoretical) question is raised withregard to computational partial differential equations: is it possible to construct a functionaliterative algorithm for the regularized long-wave (RLW) equation (also termed theBenjamin–Bona–Mahony equation) [1]
ηt + c0ηx + 3c02h0
ηηx − h206
ηxxt = 0 (1.1)
through an integral equation formalism? Here, η and c0 denote wave elevation and charac-teristic velocity (of the square root of the product of gravitational acceleration g and waterdepth h0)
c0 = √gh0, (1.2)
respectively.Physically, the nonlinear partial differential equation (1.1) models a one-way wave prop-
agation of water waves over a horizontal bottom valid for (weakly) nonlinear and fairly longwaves, being widely studied and frequently used for simulating nonlinear dispersive wavesin shallow water in applied sciences and coastal engineering. Being an alternative model tothe Korteweg–de Vries (KdV) equation, the RLW (1.1) was firstly used by Peregrine for itsadvantages in numerical computations [2] and examined extensively by Benjamin et al. [3].Here, the term “regularized” is used, because of the much better nature of the dispersionrelation between frequency ωBBM and wave number k
ωBBM = c0k
1 + h20k2
6
, (1.3)
when compared to that of the KdV equation. This is the main reason why (1.1) is much morereferred for a one-way propagation model.
There exists only few about the analytical studies on the nonlinear equation (1.1); e.g.,Benjamin et al. [3] found the analytical solution for the equation just under the restricted initialand boundary conditions [3]. Thus, finding its numerical solutions is of practical importanceand the availability of accurate and efficient numerical methods is essential [4]. Over the lastfew decades, significant efforts have been made for developing useful numerical proceduresfor (1.1). The finite difference approachwould be themost typical numerical scheme, approx-imating (1.1) with difference equations. For instance, Eilbeck and McGuire [5] investigatednumerical solitary wave solutions of the RLW equation to show that they exhibit true solitonbehavior, being stable on collision with other solitary waves [5]. Bhardwaj and Shankar [6]employed quintic spline technique and splitting method to develop a new-finite differencemethod, which is used to model solitary wave motion and undular bore development [6].Cai [7] derived a 6-point multisymplectic Preissman scheme from its Bridges’ multisym-plectic form, where backward error analysis is implemented and the performance and theefficiency of the new scheme are illustrated by solving several test examples [7]. However,finite element schemes also have been one of the most commonly used numerical methods,which are extensively employed in the area of computational partial differential equations.Challenging issues are found; for example, Gardner et al. [8] used a B-spline finite elementmethod to solve the RLW equation numerically, which is involved with a Galerkin methodwith quadratic B-spline finite elements [8]. Dag et al. [9] applied Cubic B-spline functions todevelop a collocation method for the nonlinear numerical solutions, which is used to modelsolitary waves, undular bore development and wave generation [9].
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Nowwe return to the question at the beginning about how to construct a functional iterativealgorithm of the RLW equation based on an integral equation formalism. For that, we willintroduce the notion of pseudo-parameter proposed by Jang [10], which is incorporated into(1.1), for establishing a nonlinear integral formalism involving the parameter. This enablesus to obtain dual nonlinear integral equations, equivalent to (1.1). Here, it is pointed out thatthe integral equations are not singular but regular, which inherently differ from not onlythe boundary integral equations resulted from boundary element methods but also the weakintegral formulations fromfinite elementmethods.We then apply Banach fixed point theorem[10,11] to the integral equations and derive a (required) functional iteration algorithm forintegrating (1.1), implying that the answer to the question may be found.
The iteration algorithm derived in this paper is semi-analytic and derivative-free, whichprovides a fast convergence speed for achieving highly accurate numerical solutions as isillustrated in the numerical experiment on moving solitary waves. The numerical results arealso compared to those of the conventional excellent methods. Here, it should be emphasizedthat the derived algorithm works even for a much wider range of pseudo-parameter valuesthan that of Jang [10]; e.g., the pseudo-parameter can be a complex number (rather than a realnumber), in the present study. Thus, a complex (numerical) calculation is naturally requiredfor performing the iteration, which leads to a complex numerical solution. This point is insharp contrast to the previous article by Jang [10], where pseudo-parameter values were onlyrestricted to some part of the real numbers and thus a real (not complex) numerical calculationwas required. This may be best illustrated by showing the regions of convergence (ROCs)subject to some constraints in the complex plane, presented in numerical experiment section.In particular, for a further relevant investigation for nonlinear dispersive water waves, thealgorithm is also found to be useful at observing the nonlinear behavior near a wave trainfront. Here, the nonlinear feature of the front wave is confirmed; i.e., the front wave has morepeaked crests and flatter troughs, as it becomes steep, and the wave moves faster with largeramplitude.
2 Integral Formalism
Here, we will present an integral formalism which corresponds to the RLW equation (1.1).We first modify the governing equation (1.1), following the concept of pseudo-parameter[10]. To be specific, we introduce a pseudo-parameter α ∈ C in the present study, whereC denotes the set of complex numbers, and add the product of α · η to both sides of (1.1),leading to
ηt + c0ηx − h206
ηxxt + α · η = ϕ. (2.1)
The forcing term ϕ in (2.1) involves η as well as its derivative ηx , i.e.,
ϕ(η, ηx ) ≡ − 3c02h0
ηηx + α · η. (2.2)
It immediately follows that (2.1), combined with (2.2), is still equivalent to the originalEq. (1.1), regardless of complex values of α.
The parameter α introduced above will play a crucial role in formulating an integralformalism for (1.1), which is necessary for obtaining a system of nonlinear integral equations,as shall be discussed later. We begin with applying two successive integral transforms.
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2.1 Integral Transformations
Here, associatedwith (2.1), we aim to find the successive Fourier and Laplace transforms of η.First of all, we apply Fourier integral transform to both sides of the partial differential equationof (2.1) to find an ordinary differential equation. We denote by η the Fourier transform F ofη with respect to space x as [10,12]
F(η) ≡∫ ∞
−∞η(x, t) · eikxdx ≡ η(k, t) (2.3)
for a parameter k. Then, the inverse Fourier transform F−1 is expressed as
η(x, t) = F−1(η) = 1
2π
∫ ∞
−∞η(k, t) · e−ikxdk. (2.4)
If we assume a localized wave motion of η in x , i.e.,
η, ηx → 0, as x → ±∞, (2.5)
we have the (well known) differentiation properties of Fourier transform
F(ηx ) = (−ik)η,F(ηxx ) = (−ik)2η. (2.6)
This allows (2.1) to be converted into an (first order) ordinary differential equation for η,
dη
dt+ c0(−ik)η − h20
6
d
dt
[(−ik)2η
] + αη = ϕ. (2.7)
Next, the Laplace transform L of η is to be performed on (2.7) with respect to time t ,denoted by η∗, i.e.,
L(η) ≡∫ ∞
0η(k, t) · e−st dt ≡ η∗(k, s) (2.8)
for a parameter s [10,12]. This results simply in an algebraic equation for η∗ (i.e., L [F(η)]),
(sη∗ − η1
) + c0(−ik)η∗ − h206
[(−ik)2(sη∗ − η1)
] + αη∗ = ϕ∗
or
η∗[
s
(
1 − h206
(−ik)2)
+ c0(−ik) + α
]
= ϕ∗ + η1
(
1 − h206
(−ik)2)
. (2.9)
Here, the Laplace-transform property of differentiation is employed together with initialcondition;
(dη/dt)∗ = sη∗ − η(k, 0) = sη∗ − η1, (2.10)
in which η1 designates an initial wave profile, i.e., η1(x) = η(x, 0). (2.9) is readily solvedfor η∗ to give
η∗ = 1
1 − h206 (−ik)2
·ϕ∗ + η1
(1 − h20
6 (−ik)2)
s + c0(−ik)+α
1− h206 (−ik)2
= 1
1 + h20k2
6
·ϕ∗ + η1
(1 + h20k
2
6
)
s − i (c0k−α/ i)
1+ h20k2
6
. (2.11)
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With the definition of a (complex-valued) function ωBBM such that
ωBBM(k) ≡ c0k − α/ i
1 + h20k2
6
, (2.12)
the expression (2.11) may be further shortened; (2.11) admits the partial fractions represen-tation in the simple form in terms of the ωBBM
η∗ = η1
s − iωBBM+ ωBBM
c0k − α/ i· ϕ∗
s − iωBBM. (2.13)
Remark 1 The newly defined ωBBM in (2.12) converges toωBBM in (1.3) of theRLWequation(1.1), as the pseudo-parameter α ∈ C tends to zero: i.e.,
ωBBM → ωBBM, asα → 0. (2.14)
2.2 Inverse Integral Transformations
We start with taking the inverse Laplace transform L−1 on the first term of the right handside of (2.13), which is
L−1[
η1(k)
s − iωBBM
]= η1(k) · L−1
[1
s − iωBBM
]
= η1(k) · [exp (iωBBM · t)] (2.15)
because 1/(s − a) = L[exp(at)
]for a ∈ C from table [10,12]. Performing further the
inverse Fourier transform F−1 on the above result, being denoted by I1, would lead to
I1(x, t) ≡ F−1 {η1(k) · [
exp (iωBBM(k) · t)]}
= {F−1 [η1(k)]
} ∗ {F−1 [
exp (iωBBM(k) · t)]} (2.16)
by Fourier convolution theorem, where the notation ∗ means Fourier convolution.I1 in (2.16) can be calculated as
I1 = {η1(x)} ∗{
1
2π
∫ ∞
−∞exp
[iωBBM(k) · t] · exp(−ikx)dk
}
= {η1(x)} ∗{
1
2π
∫ ∞
−∞e−i[kx−ωBBM(k)·t]dk
}
= 1
2π
∫ ∞
−∞
∫ ∞
−∞e−i[k(x−ξ)−ωBBM(k)·t] · η1 (ξ) dξdk, (2.17)
owing to the formula of inverse Fourier transform (2.4), definition of Fourier convolutionand the identities
F−1η1 = (F−1F)η1 = η1(x). (2.18)
If we define the (complex) pseudo-phase function
θ (x, t; k) ≡ kx − ωBBM(k) · t. (2.19)
(2.17) then may be simplified into
I1(x, t) = 1
2π
∫ ∞
−∞η1(k)·e−i θ (x,t;k)dk, (2.20)
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which describes physically the superposition of (complex) elementary wave solutionse−i θ (x,t;k) with a wave spectrum η1, i.e., the Fourier transform of the initial profile η1(x);
η1(k) = F(η1(x)) =∫ ∞
−∞η1 (ξ) · eikξdξ . (2.21)
Next, we move on to the inverse Laplace transform L−1 on the second term of (2.13),
L−1[
ωBBM
c0k − α/ i· ϕ∗
s − iωBBM
]= ωBBM
c0k − α/ i· L−1
[1
s − iωBBM· Lϕ
]
= ωBBM
c0k − α/ i· L−1 [
L[exp (iωBBM(k) · t)] · Lϕ
]
= ωBBM
c0k − α/ i· exp [
iωBBM(k) · t] ◦ ϕ
= ωBBM
c0k − α/ i
∫ t
0exp
[iωBBM(k) · (t − τ)
] · ϕ(k, τ )dτ,
(2.22)
where the formula 1/(s − a) = L[exp(at)
]for a ∈ C and Laplace convolution theorem are
used [10,12]. And we take the inverse Fourier transform F−1 on the above result, giving atriple integral I2,
I2(x, t) ≡ F−1{
ωBBM
c0k − α/ i
∫ t
0exp
[iωBBM(k) · (t − τ)
] · ϕ(k, τ )dτ
}
= 1
2π
∫ ∞
−∞
{ωBBM
c0k − α/ i·∫ t
0exp
[iωBBM(k) · (t − τ)
]
·(∫ ∞
−∞ϕ (ξ, τ ) · eikξdξ
)dτ
}· e−ikxdk
= 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
exp[iωBBM(k) · (t − τ)
]
· eik(ξ−x) · ϕ (ξ, τ ) dξdkdτ
= 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· e−i θ (x−ξ,t−τ ;k) · ϕ (ξ, τ ) dξdkdτ (2.23)
by the use of Fourier formula (2.3), its inverse (2.4) and (2.12).
Remark 2 The integral I2 of (2.23) can be also expressed as a single integral in the form
I2(x, t) = 1
2π
∫ t
0ζ(x, t, τ ;ϕ)dτ . (2.24)
Then, the integrand ζ physically represents the superposition of elementary wave solutionse−i θ (x,t−τ ;k) with time shift τ , which has awave spectrum ϕ(k, τ )/(1+h20k
2/6) parametrizedby the τ :
ζ(x, t, τ ;ϕ) ≡∫ ∞
−∞ϕ(k, τ )
1 + h20k2
6
· e−i θ (x,t−τ ;k)dk (2.25)
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2.3 Establishing Integral Formalism
This subsection is devoted to the realization of an integral formalism from the results in theprevious subsection. This is simply achieved by recalling that η is recovered by taking thetwo successive inverse Fourier and Laplace transforms on both sides of (2.13), i.e., for apseudo-parameter α ∈ C, η is symbolically represented as
η = (F−1L−1)η∗ = I1 + I2
= 1
2π
∫ ∞
−∞η1(k) · e−i θ (x,t;k)dk
+ 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· e−i θ (x−ξ,t−τ ;k)ϕ (ξ, τ ) dξdkdτ (2.26)
from (2.20) and (2.23). The relation (2.26) may be considered as our integral formalismwhich corresponds to (1.1), where ϕ is directly related with η and its derivative through (2.2).
3 Derivation of Functional Iteration Algorithm
In this section, we shall derive a new functional iteration algorithm for solving the RLWequation (1.1). To this end, (1.1) is transformed into a system of nonlinear integral equationsby using the results of the previous section. And then, we employBanach contraction theorem[10,11], applied to the system of integral equations, which makes us derive an (numerical)iterative strategy (i.e., a functional iteration algorithm for (1.1)).
3.1 Dual Integral Equations
With the integral formalism (2.26) established in Sect. 2.3, we will obtain nonlinear integralequations. For that, let us substitute (2.2) into (2.26) to find
η(x, t) = 1
2π
∫ ∞
−∞η1(k) · e−i θ (x,t;k)dk
− 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞
3c02h0
1 + h20k2
6
· e−i θ (x−ξ,t−τ ;k) · η(ξ, τ ) · ψ(ξ, τ )dξdkdτ
+ α
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· e−i θ (x−ξ,t−τ ;k) · η(ξ, τ )dξdkdτ (3.1)
where ψ indicates a new variable, corresponding physically to wave slope ηx . And, wedifferentiate the above with respect to x , giving
ψ(x, t) = 1
2π
∫ ∞
−∞η1(k) · ∂
∂xe−i θ (x,t;k)dk
− 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞
3c02h0
1 + h20k2
6
· ∂
∂xe−i θ (x−ξ,t−τ ;k) · η(ξ, τ ) · ψ(ξ, τ )dξdkdτ
+ α
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· ∂
∂xe−i θ (x−ξ,t−τ ;k) · η(ξ, τ )dξdkdτ . (3.2)
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The pair of (3.1) and (3.2) then constitutes two coupled nonlinear integral equations of secondkind for the two unknown functions of η and ψ .
The integral equations of the form (3.1) and (3.2) may be abbreviated, with some pre-liminaries of background material of function spaces. Denote by BT the collection ofcomplex-valued continuous and bounded functions defined on R × [0, T ] for T > 0, whereR is the set of real numbers. Analogously, let B0 be the collection of real-valued continuousand bounded functions defined on R. Then, associated with (3.1) and (3.2), we can define anintegral operator J : B0 → BT such that
J[η1](x, t) ≡ 1
2π
∫ ∞
−∞η1(k)·e−i θ (x,t;k)dk (3.3)
for any η1 ∈ B0. In a similar way, we define K : BT → BT such that
K[η](x, t) ≡ 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
e−i θ (x−ξ,t−τ ;k) · η(ξ, τ )dξdkdτ (3.4)
for any η ∈ BT .By virtue of the integral operators J in (3.3) and K in (3.4), we can write (3.1) as
η = J[η1] − 3c02h0
K[η · ψ] + αK[η] (3.5)
and (3.2) as
ψ = Jx [η1] − 3c02h0
Kx [η · ψ] + αKx [η]. (3.6)
The two operator equations of (3.5) and (3.6) can be considered as an abbreviated form forthe integral equations described by (3.1) and (3.2). Here, the new operators Jx and Kx havethe expressions
Jx [η1](x, t) ≡ 1
2π
∫ ∞
−∞η1(k)· ∂
∂xe−i θ (x,t;k)dk for η1 ∈ B0 (3.7)
and
Kx [η](x, t) ≡ 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· ∂
∂xe−i θ (x−ξ,t−τ ;k) · η(ξ, τ )dξdkdτ
for η ∈ BT , (3.8)
respectively.
3.2 Deriving Functional Iterative Algorithm
In this subsection, we present the derivation of a functional iteration algorithm to solve (1.1)by means of the integral equations obtained in the previous subsection.
Let us define a (integral) nonlinear operator
T[η,ψ] ≡ J[η1] − 3c02h0
K[η · ψ] + αK[η] (3.9)
for any η, ψ ∈ BT , so that (3.5) can be more shortened into
η = T[η,ψ]. (3.10)
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Then, the integral equations (3.1) and (3.2) are much more abbreviated as
η = T[η]. (3.11)
Here, the nonlinear operator T has the form (the superscript t denotes the transpose)
T[η] ≡ (T[η,ψ],Tx [η,ψ])t (3.12)
in terms of a new variable η = (η, ψ)t , in which the operator Tx stands for the right handside of (3.6), i.e.,
Tx [η,ψ] = Jx [η1] − 3c02h0
Kx [η · ψ] + αKx [η]. (3.13)
Based on the fact that η is invariant under the mapping T in (3.11), we would like topropose a functional iteration algorithm for ηn ≡ (ηn, ψn)
t , where ηn, ψn ∈ BT ,
ηn+1 = T[ηn] for n = 0, 1, 2, ... (3.14)
with the initial guessη0 ≡ (0, 0)t (3.15)
by appealing to Banach fixed point theorem [10,11]. That is, (3.14) is a recurrence relationwhich recursively defines a functional sequence
{ηn
}, n = 0, 1, 2, ....
Remark 3 Being semi-analytic and derivative-free iterative strategy, the iteration (3.14)yields a solution η of (1.1) as well as its derivative ηx (= ψ) as a bonus, as the numberof iteration n increases, provided that (3.14) converges.
We recall that the model (1.1) physically describes a one-way wave propagation of non-linear dispersive waves η(x, t) over a horizontal bottom as discussed in Introduction, whereη(x, t) is real-valued, if a real-valued initial condition η1(x) = η(x, 0) is taken into account.Suppose that η is a (real-valued) wave solution of the initial value problem for (1.1) withthe real-valued initial condition. Then, the η should also satisfy the α-parameterized partialdifferential equations of (2.1) and (2.2) together with the initial condition, for any pseudo-parameter α ∈ C, due to the fact that (2.1) and (2.2), for α ∈ C, are equivalent to (1.1)as mentioned at the beginning of Sect. 2. And, the η even further should satisfy the systemof integral equations (3.1) and (3.2), or alternatively, the integral operator equation (3.11),because (3.1) and (3.2) describe an (equivalent) integral equation formulation for (1.1) com-bined with the initial condition.
If T in the integral operator equation (3.11) has a contractive nature, the η then may beexpressed as a limit of a sequence {ηn}, n = 0, 1, 2, ..., generated by (3.14) (recalling thatη is invariant under T) and the sequence converges (See Banach fixed point theorem [11]).The sequence is a Cauchy sequence, which has a unique limit in a Banach space (such asour solution space), because every Banach space is topologically a Hausdorff space (Roman[11]); i.e.,
ηn → η as n → ∞. (3.16)
Remark 4 (3.16) indicates that a limit of a sequence {ηn}, n = 0, 1, 2, ...by (3.14) is uniqueand real-valued, if a real-valued initial condition η1(x) = η(x, 0) is imposed. However, if nis finite, the output ηn may be complex-valued, whose real part converges to the true solutionof (1.1) as n → ∞, while its imaginary part goes to zero. That is,
Re (ηn) → η and Im (ηn) → 0 as n → ∞, (3.17)
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followed by the fact that ηn = Re (ηn) + i · Im (ηn) → η as n → ∞ by (3.16) and the limitη is real-valued.
3.3 Iterative Algorithm for the Special Case α = 0
Notice that it is a complex value that the pseudo-parameter α of the iteration (3.14) involves.This demands that the output of the numerical iteration (3.14) should be complex-valued fora finite n. However, if we set α = 0 in (3.14), the output becomes real-valued.
Suppose that the pseudo-parameter α ∈ C vanishes in (3.14). Then, I1 in (2.17) or (2.20)reduces to
(I1)α=0 = 1
2π
∫ ∞
−∞
∫ ∞
−∞e−i[k(x−ξ)−ωBBM(k)·t] · η1 (ξ) dξdk
= 1
π
∫ ∞
0
∫ ∞
−∞cos [θ(x − ξ, t; k)] · η1 (ξ) dξdk (3.18)
where θ represents the phase function θ in (2.19) when α = 0, i.e.,
Here, the kernel in (3.18) results from the Euler formula
e−iθ = cos θ − i sin θ (3.20)
and the anti-symmetric property
sin [θ(x − ξ, t;−k)] = sin [−θ(x − ξ, t; k)]= − sin [θ(x − ξ, t; k)] (3.21)
because ωBBM and (thus) θ are odd in k.Similarly, from (2.23)
(I2)α=0 = 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· e−iθ(x−ξ,t−τ ;k) · ϕ (ξ, τ ) dξdkdτ
= 1
π
∫ t
0
∫ ∞
0
∫ ∞
−∞1
1 + h20k2
6
cos [θ(x − ξ, t − τ ; k)]ϕ (ξ, τ ) dξdkdτ , (3.22)
because of the anti-symmetry (in k)
sin [θ(x − ξ, t − τ ;−k)] = sin [−θ(x − ξ, t − τ ; k)]= − sin [θ(x − ξ, t − τ ; k)] . (3.23)
Therefore, (3.14) becomes
ηn+1(x, t) = 1
π
∫ ∞
0
∫ ∞
−∞cos [θ(x − ξ, t; k)] · η1 (ξ) dξdk
− 1
π
∫ t
0
∫ ∞
0
∫ ∞
−∞
3c02h0
1 + h20k2
6
· cos [θ(x − ξ, t − τ ; k)]
·ηn(ξ, τ ) · ψn(ξ, τ )dξdkdτ (3.24)
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and
ψn+1(x, t) = 1
π
∫ ∞
0
∫ ∞
−∞∂
∂xcos [θ(x − ξ, t; k)] · η1 (ξ) dξdk
− 1
π
∫ t
0
∫ ∞
0
∫ ∞
−∞
3c02h0
1 + h20k2
6
· ∂
∂xcos [θ(x − ξ, t − τ ; k)]
·ηn(ξ, τ ) · ψn(ξ, τ )dξdkdτ (3.25)
for n = 0, 1, 2, ..., respectively, due to (3.18) and (3.22), when α = 0; with emphasis onthat I1 in (2.17) and I2 in (2.23) remain pure real. It is clear that the functional iterationconsisting of (3.24) and (3.25) derived above is a recurrence relation which does not involvea pseudo-parameter α ∈ C at all (i.e., it is pseudo-parameter free.) and thus both ηn andψn are real-valued. That is, the output of the iteration of (3.24) and (3.25) is real-valued, asmentioned above in this subsection.
Remark 5 As a special case of (3.14), the pair of (3.24) and (3.25), which corresponds toα = 0, is the simplest (or the most efficient) recurrence relation among our α-parameterfamily of (3.14); in fact, the pair is an iterative strategy whose output is real-valued. Eventhough the pair is the simplest, it produces the highest levels of accuracy in the presentiterative strategy (3.14), as can be seen later (see Sect. 4).
4 Numerical Experiments
4.1 Numerical Performance
This subsection concerns numerical experiments performed on solitarywave propagations toexaminewhether the functional iteration algorithm (3.14)works. Thereby, wewill investigatethe effect of the pseudo-parameter α ∈ C introduced in Sect. 2 on the numerical performanceof the iteration (e.g., the iteration’s convergence speed and accuracy).
We first introduce three appropriate dimensionless variables, defined by
η∗ = 3η
2h0, t∗ = t
√6g
h0, x∗ =
√6
h0x . (4.1)
Then, the (original) physical partial differential equation (1.1) is transformed into a dimen-sionless RLW equation [13] (see “Appendix A”)
η∗t∗ + η∗
x∗ + η∗η∗x∗ − η∗
x∗x∗t∗ = 0. (4.2)
It is well known that (4.2) allows a solitary wave (or soliton) moving to the right in the form[13]
η∗ex
(x∗, t∗
) = 3d · sech2 [κ
(x∗ − vt∗ − x∗
0
)], (4.3)
where 3d indicates dimensionless amplitude, v wave velocity(= d + 1), and 2π /κ wavewidth:
κ = √d/4v (4.4)
In original physical variables, (4.3) may be re-expressed as
ηex (x, t) = 2h0d · sech2[
κ√6
h0
(x − v
√gh0t − x0
)]
. (4.5)
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By iterating the recursion of (3.14) together with the (zero) initial guess (3.15), we nownumerically simulate the solitary wave of (4.3) or (4.5) using the initial condition imposedby
η∗1
(x∗) = η∗ (
x∗, 0) = 3d · sech2 [
κ(x∗ − x∗
0
)](4.6)
or
η1 (x) = η (x, 0) = 2h0d · sech2[
κ√6
h0(x − x0)
]
. (4.7)
Recall that (3.14) involves the pseudo-parameter α which is in general complex. For thisreason, a complex domain calculation is generally demanded for the numerical simulationas was discussed before. However, we start with the simplest case of (3.24) and (3.25) (whenα = 0).
4.1.1 Simulations for α = 0
We iterate the pair of (3.24) and (3.25), corresponding to caseα = 0, with the initial condition(4.6) or (4.7) to obtain the solitary wave solution of (4.3) or (4.5). For the simulation, thedimensionless amplitude 3d is chosen as 0.3, and gravitational acceleration g andwater depthh0 as 9.80665m/s2 and 1.0 m, respectively. Here, the usual Simpson integration rule is usedto estimate the integrals appearing in the pair (but, with the trapezoidal rule in time), wherewe discretize dimensionless temporal interval 0 < t∗ < T into (N − 1) equal length-parts�t∗ = T/(N−1) and dimensionless spatial interval a < x∗ < b into (M−1) equally spacedpanels �x∗ = (b − a)/(M − 1), giving x∗
i (these points will be used for error estimationlater) for positive integers i (1 ≤ i ≤ M),
x∗i = a + (i − 1)
(b − a)
M − 1. (4.8)
Figures 1 and 2 show the numerical plot of the solitary wave (4.3) and its numericalconvergence behavior, respectively, where dimensionless spatial and temporal computationaldomains are taken as −40 < x∗ < 60 and 0 < t∗ < 20 respectively, with their incrementsof �x∗ = 1/8 and �t∗ = 0.1; i.e., a = −40, b = 60, T = 20, M = 801, N = 201. Areasonable wave solution can be observed by only the first iteration when t∗ is small and thereappears to be an almost converged solution valid for the whole time interval when n = 12, asdemonstrated in Fig. 2. With dimensionless time fixed (t∗ = 20), the numerical convergencebehavior is depicted in Fig. 3.
Fig. 1 Plot of the movingsolitary wave (4.3)
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Fig. 2 Convergence behavior of the proposed iteration for the solitary wave solution (4.3): α = 0,�x∗ = 1/8and �t∗ = 0.1
Fig. 4 Two errors of Err2 in (4.9) and Err∞ in (4.10) at t∗ = 10, 20: α = 0, �x∗ = 1/8 and �t∗ = 0.1
Figure 4 illustrates that the discrete root mean square norm Err2 and the maximum errorErr∞[14] decay rapidly as the number of iterations n increases;
Err2(t∗) = ∥∥η∗
ex(x∗, t∗) − η∗
n(x∗, t∗)
∥∥2 =
√√√√�x∗ ·M∑
i=1
∣∣η∗ex(x
∗i , t∗) − η∗
n(x∗i , t∗)
∣∣2 (4.9)
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Table 1 Numerical values given by the proposed iteration for Err2 in (4.9), Err∞ in (4.10) and invariantsin (4.11–4.13) for various dimensionless times: α = 0, �x∗ = 1/8, �t∗ = 0.1
Time t∗ Err2 × 103 Err∞ × 103 C1 C2 C3
Analytical 0 0 3.97993 0.81046 2.57901
0 0 0 3.97993 0.81046 2.57901
4 0.007657 0.005538 3.97993 0.81046 2.57901
8 0.008472 0.003726 3.97992 0.81046 2.57901
12 0.008711 0.003367 3.97993 0.81046 2.57901
16 0.008779 0.003102 3.97992 0.81046 2.57901
20 0.008929 0.002902 3.97988 0.81046 2.57901
Table 2 Numerical values given by the proposed iteration and other conventional excellent methods for Err2in (4.9), Err∞ in (4.10) and invariants in (4.11–4.13) at specified dimensionless time t∗ = 20: α = 0
PG FEM [13] 0.06493 0.02643 3.97990 0.81050 2.57900
SQBS CM [20] 0.04315 0.01321 3.97989 0.81050 2.57900
and
Err∞(t∗
) = ∥∥η∗ex
(x∗, t∗
) − η∗n
(x∗, t∗
)∥∥∞ = maxi=1,..,M
∣∣η∗ex
(x∗i , t∗
) − η∗n
(x∗i , t∗
)∣∣ ,
(4.10)where two norms ‖·‖2 and ‖·‖∞ are taken over the x∗ variable only, with dimensionlesstime t∗ being kept fixed (t∗ = 10, 20). The values of errors are tabulated in Table 1 forvarious specified dimensionless times. Note that the order of accuracy is of 10−6, beingaccurate. They are also compared to other results of the conventional excellent methods as isTable 2.
It may be worth estimating numerical values for the following three conservative (invari-ant) quantities corresponding to mass, momentum and energy [14,15]:
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C1(t∗
) =∫ 60
−40η∗n
(x∗, t∗
)dx∗ � �x∗
M∑
i=1
η∗n
(x∗i , t∗
), (4.11)
C2(t∗
) =∫ 60
−40η∗2n
(x∗, t∗
) +(
∂η∗n
∂x∗
)2 (x∗, t∗
)dx∗
� �x∗M∑
i=1
{
η∗2n
(x∗i , t∗
) +(
∂η∗n
∂x∗
)2 (x∗i , t∗
)}
, (4.12)
and
C3(t∗
) =∫ 60
−40η∗3n
(x∗, t∗
) + 3η∗2n
(x∗, t∗
)dx∗ � �x∗
M∑
i=1
{η∗3n
(x∗i , t∗
) + 3η∗2n
(x∗i , t∗
)},
(4.13)
where, we review that (1.1) is known to have exactly the three conservation laws (4.11–4.13). Being compared to the analytical results, the numerical values of the invariants of C1,C2, C3 in (4.11–4.13) by the proposed iteration of (3.24) and (3.25) are shown for variousdimensionless times in Table 1. For other two types of increments, e.g., �x∗ = �t∗ = 0.50and �x∗ = �t∗ = 0.25, the corresponding errors and invariants are presented as well inTable 2.
4.1.2 Simulations for α Other than Zero
In Sect. 4.1.1, we examined the special case of the iterative algorithm (3.14), i.e., the pairof (3.24) and (3.25) which corresponds to (3.14) when α = 0, for the numerical experimenton a solitary wave. However, in the present subsection, we extend the pseudo-parameterα to complex numbers (of course, which include real numbers) to explore its effect on thenumerical results. Using the same integration numerical rules as was employed in Sect. 4.1.1,weperform (3.14) togetherwith the initial condition (4.6) or (4.7) to simulate again the solitarywave solution of (4.3) or (4.5). Here, the dimensionless computational domains are chosen tobe−35 < x∗ < 35 and 0 < t∗ < 4 respectively, but with the larger increments of�x∗ = 0.5and �t∗ = 0.2 (than before).
It should be noted that, for non-zero complex numbers α, the iterative solutions ηn(x, t)for finite integers n produced by (3.14) are in general complex-valued (as was pointed outin Sect. 3.2), because the (two) concerning operators J in (3.3) and K in (3.4) are complex-valued. So, generally, the real parts of iterative solutions converge to a true solution as thenumber of iteration n increases, while the imaginary parts of iterative solutions converge tozero (See Remark 4).
However, we find it interesting to observe an exceptional case, where the iterative solutionsηn(x, t) for finite integers n produced by (3.14) are proved to be all real-valued (not complex-valued), especially when α is pure real (because the iteration (3.14) becomes essentially areal-valued (recurrence) equation if α is real. See “Appendix B”). Bearing in mind this, theiterative algorithm (3.14) yields Fig. 5, which exhibits only convergence behaviors of the realparts of ηn(x, t), when the pseudo-parameter is selected to be pure real; α = 1. Here, thereal parts of the iterative solutions converge to a true solution as the number of iteration nincreases.
Figures 6 and 7 present the convergence characteristics of (3.14) for the pure imaginarycase, α = i . The convergence behavior (real part) of Fig. 6 is analogous to that of the
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Fig. 5 Convergence behavior (real part) of the proposed iteration for the solitary wave solution (4.3): α = 1,�x∗ = 0.5, �t∗ = 0.2
Fig. 6 Convergence behavior (real part) of the proposed iteration for the solitary wave solution (4.3): α = i ,�x∗ = 0.5, �t∗ = 0.2
previous case of α = 1 in Fig. 5, and the imaginary parts of the iterative solutions are seento converge to zero in Fig. 7. Figures 8 and 9 account for the nature of solution convergencefor α =1+i . Figures 10, 11 and 12 plot the iterative solitary wave profiles with dimensionlesstime kept fixed (t∗=4). Here, note again that Fig. 10 only shows convergence behaviors of thereal parts, because the iterative solutions are real-valued with α pure real, as was discussedabove.
As was discussed earlier, the pseudo-parameter α was artificially introduced at the begin-ning of this study and it may be crucial in that it affects the nature of convergence andaccuracy of the iteration (3.14) derived (or proposed) in this paper. Indeed, Fig. 13 revealsthe characteristics of convergence and accuracy depending on the parameter α for a fixeddimensionless time t∗ = 4, where the real and imaginary parts of errors are defined as
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Fig. 7 Convergence behavior (imaginary part) of the proposed iteration for the solitary wave solution (4.3):α = i , �x∗ = 0.5, �t∗ = 0.2
Fig. 8 Convergence behavior (real part) of the proposed iteration for the solitarywave solution (4.3):α = 1+i ,�x∗ = 0.5, �t∗ = 0.2
(Err2)real = ∥∥η∗ex − Re(η∗
n)∥∥2 , (4.14)
(Err2)imag = ∥∥0 − Im(η∗n)
∥∥2 . (4.15)
Here, η∗ex is the exact solution of the moving solitary wave of (4.3). It is interesting to see
that the smaller the norm of α tends to contribute to the faster iteration’s convergence andmore accurate solutions.
4.1.3 Region of α ∈ C Satisfying Requirement
We have, so far, performed the numerical simulation of a solitary wave to find the numericalperformances of the algorithm, when given a specified α. However, in this subsection, we
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Fig. 9 Convergence behavior (imaginary part) of the proposed iteration for the solitary wave solution (4.3):α = 1 + i , �x∗ = 0.5, �t∗ = 0.2
Fig. 10 Convergence behaviors(real part) of the proposediteration for the solitary wavesolution (4.3) whent∗ = 4 : α = 1,�x∗ = 0.5,�t∗ = 0.2
Fig. 11 Convergence behaviors (real and imaginary parts) of the proposed iteration for the solitary wavesolution (4.3) when t∗ = 4 : α = i, �x∗ = 0.5, �t∗ = 0.2
would like to look for inversely pseudo-parameters α which satisfy a (planned) performancerequirement of the algorithm through the numerical simulation. In practice, this would be anissue of clear engineering importance.
With the use of the same computational condition as that of Sect. 4.1.2, Fig. 14 depictsthe two shaded regions of �1 and �2 of pseudo-parameters α ∈ C satisfying (cf. see (4.14))
∥∥η∗ex − Re(η∗
n)∥∥2 < ε, (4.16)
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Fig. 12 Convergence behaviors (real and imaginary parts) of the proposed iteration for the solitary wavesolution (4.3) when t∗ = 4 : α = 1 + i , �x∗ = 0.5,�t∗ = 0.2
Fig. 13 Convergence characteristics of the proposed iteration against n depending on α, when t∗ = 4 :�x∗ = 0.5,�t∗ = 0.2
which is an inequality of a performance requirement with a tolerance ε. The two regions(ε = 0.1), plotted in the complex planeC as in Fig. 14a, b, respectively, both look symmetricwith respect to the real axis R and similar. In addition, the former is bigger than the latter.This is because the algorithm produces a less accurate solution with higher dimensionlessamplitude, when given a computational (discretization) condition; therefore, ultimately, there
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Fig. 14 Two shaded regions of pseudo-parameters α satisfying the performance requirement (4.16) withtolerance ε = 0.1; Specifically, for a dimensionless amplitude 3d1 = 0.3, an enlarged rectangle is inserted,containing 10 red-colored points which correspond to the pseudo-parameters α used in Sect. 4.1.2 Similarly,an enlarged rectangle is inserted as well for b dimensionless amplitude 3d2 = 1.2, where the 10 red-coloredpoints are also depicted
Fig. 15 Convergence characteristics of the proposed iteration where the pseudo-parameters α lie in the realaxis R but inside the shaded regions of �1 and �2 in Fig. 14; In particular, for a 3d1 = 0.3 and b 3d2 = 1.2,the points of α = 13.5 and α = 9.5 lie on the boundaries of �1 and �2, respectively
may exist an empty space which satisfies (4.16) for a (very) high dimensionless amplitude.Here, it is also observed that the α ∈ C, considered for the numerical experiment in the abovesubsections, are inside the two regions (i.e., α ∈ �1,�2), which, in fact, are located near theorigin in the complex plane C (see the enlarged images in Fig. 14).
The regions can be also considered as the ROCs subject to the constraint (4.16), due tothe fact that they clearly show parts of ROCs of the present iterative algorithm. If α werelimited to real numbers as was done in Jang [10], then, the corresponding ROCs subject tothe constraint (4.16) would be the intersections of �1 and �2 with the real axis R. Because�1 and �2 are much larger than the intersections, the present algorithm is shown to havemuch wider ranges of the ROCs subject to the constraint (4.16) than those of Jang [10].
It would be instructive to investigate convergence characteristics of the proposed iteration,when α lies (very) close to the boundaries of the shaded regions. Figure 15 gives the con-vergence characteristics corresponding to α which remain strictly real but inside the shadedregions (including the boundaries). As was pointed out in Sect. 4.1.2, we also observe that thesmaller the (real) values of α contribute to the faster convergence and more accurate results.
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4.2 Nonlinear Behavior Near Wave Train Front
In this subsection, we will explore the nonlinear behavior near the front of a slowly varyingwave train due to a local disturbance by using the present algorithm. This is a fascinatingmatter in water waves, in fact, which has been significantly examined in the past (e.g., inrelation with tsunami). However, the classical (theoretical) study was restricted essentially tothe linear approximation; cf. see Kajiura [21] who found the leading wave (linear) solutionin the form of an Airy function [21].
For simulating a wave train, we first consider an initial value problem of the governingequation (1.1) and the initial conditions
η∗1 = Ai · exp [−Bx∗2] , i = 1, 2, 3, (4.17)
or, alternatively,
η1 = 2h0Ai
3· exp
[
−6B
h20x2
]
, i = 1, 2, 3, (4.18)
for constants Ai > 0 and B > 0, which are known as the Maxwellian conditions [22]. Andthen, the algorithm (3.24) and (3.25), which corresponds to α = 0, is iterated to solve theabove initial value problem for three different Ai > 0, i.e., A1 = 0.01, A2 = 0.20 andA3 = 0.27, where the concerning computational domains are selected as 20m < x < 140m(M = 481,�x = 1/3m) and 0sec< t < 35sec (N = 211,�t = 1/6 sec); equiva-lently, -49< x∗ < 343(�x∗ ≈ 0.82), 0< t∗ < T = 268(�t∗ ≈ 1.28) in dimensionlessform.
Figure 16 indicates a typical convergence behavior of the algorithm, say for A3 = 0.27,in which, similarly as in the previous subsection (e.g., in Fig. 2), qualitatively reasonableiterative solutions η∗
n are found to be achieved at the early stage of iteration. Furthermore,it is clearly seen that the Maxwellian profile (4.17) propagates into space and time in adispersive manner.
We will next determine where to stop the (iterative) algorithm (3.24) and (3.25), based onthe norm difference e between η∗
n+1 and η∗n in this study; we would like to stop the iteration
Fig. 16 Typical convergence behavior of the proposed iteration (3.24) and (3.25) for a slowly varying wavetrain due to the Maxwellian initial condition; A3 = 0.27, B = 0.1
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Fig. 17 Plots of e in (4.19) against iteration n for three different cases of Ai > 0(i = 1, 2, 3); B = 0.1
of the algorithm, if e satisfies the inequality
e = ∥∥η∗n+1(x
∗, T ) − η∗n(x
∗, T )∥∥2 < δ (4.19)
with a (tolerance) bound δ and an end time T in numerical experiment (= 268, see the (just)above). Figure 17 uncovers plots of e against iteration n for Ai > 0, i = 1, 2, 3, whichillustrates that smaller amplitudes Ai make e decay much more rapidly, as n increases.
If we set, say δ = O(10−9) in (4.19), we then stop the iteration algorithm at n = 6 forA1 = 0.01, because e has order O(10−9) at n = 6 for A1 = 0.01 as shown in Fig. 17.Similarly, we stop the iteration at n = 25 for A2 = 0.20 and at n = 32 for A3 = 0.27,respectively, where e = O(10−9) for both cases A2 and A3 (see Fig. 17). These resultsimply that our numerically converged solutions become η∗
6, η∗25 and η∗
32 for A1 = 0.01, forA2 = 0.20 and A3 = 0.27, respectively, if δ = O(10−9) is chosen for (4.19).
Remark 6 No analytical solution of the present initial value problem has been found, sothere may be no way to estimate quantitative errors of the iterative solutions such as (4.9)and (4.10).
Figure 18 shows the numerically converged solutions of the η∗6, η∗
25 and η∗32 for three
different amplitudes, Ai > 0, i = 1, 2, 3. These describe physically the development ofnonlinear wave train fronts. Here, we clearly observe the nonlinear behaviors near wave train
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Fig. 18 Development of slowly varying wave train fronts caused by the Maxwellian conditions (4.17); Ai >
0, (i = 1, 2, 3), B = 0.1
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fronts (within blue-colored rectangles), which are compared to the linear result,
ηlinear(x, t) = 1
2π
∫ ∞
−∞η1(k) · e−i θ (x,t;k)dk. (4.20)
ηlinear in (4.20) denotes the closed form of solution for the linearized version of the RLWequation (1.1), being resulted from (2.13) when ϕ∗ vanishes.
It is found that the nonlinear behavior of the leading wave (or the wave front) is in a goodagreement with the linear one, when the smallest amplitude, i.e., A1 = 0.01, of (4.17) isconsidered, as expected. However, there is a sensible discrepancy between them for the otheramplitudes. Especially, the nonlinear feature of the leading wave (or the wave front withinblue-colored rectangles) is most revealed for the case A3 = 0.27, where we can find steepwaves with peaked crests and flat troughs like a cnoidal wave. And we further discover theinteresting nonlinear phenomenon that high-amplitude waves move faster than waves of lowamplitude.
5 Concluding Remarks
Thebasic ideaunderlying this papermaybe characterizedby thequestionwhich concerns howto construct a functional iterative algorithm for the RLW equation via an integral equationformalism. For resolving the question, we have introduced the pseudo-parameter concept[10], merged into the original RLW equation. This provides the (pseudo-parameterized) dualregular integral equations, making us derive a recurrence relation using fixed point iteration.It is a (required) new functional iterative algorithm for integrating the RLW equation.
The functional iterative algorithm derived in this paper is semi-analytic and derivative-free, which is straightforward to apply in that the only thing required is just the use of regularnumerical integration in every iterative process (because the concerning integral equations areregular). For checking the algorithm’s performance, we have made a numerical experimentwith solitary wave propagations to examine not only the convergence properties and accuracycharacteristics but also the ROCs subject to some constraints in the complex plane. Notably,the algorithm is also shown to be good at investigating the evolution of a slowly varyng nonlin-ear wave train and the nonlinear behavior near the front of the wave train, which demonstratesfundamental and interesting physical phenomena of nonlinear dispersive water waves.
It is noted that the pseudo-parameter, artificially introduced here, may be regarded as animportant parameter, which has several roles. For the first, it connects the original RLW equa-tion with the dual integral equations. Second, it is related with the iteration’s performance,e.g., the convergence speed and accuracy of the iteration algorithm. Finally, we would like toclose this paper by stressing the point that the values of the pseudo-parameter in this paperextend over a much wider range than those in the previous study in Jang [10]. This maybe clearly confirmed by presenting diagrams showing the ROCs subject to some constraintsplotted in the complex plane.
Acknowledgements This research was supported by Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01058542).And this work was also supported by the National Research Foundation of Korea(NRF) Grant funded by theKorean Government (MSIP)(NRF-2017R1A5A1015722). And, the author wishes to thank Mr. Jinsoo Parkfor his assistance.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and
reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons license, and indicate if changes were made.
Appendix A: Derivation of (4.2)
With the chain differentiation rule
∂
∂t= ∂
∂t∗dt∗
dt,
∂
∂x= ∂
∂x∗dx∗
dx
we have
∂
∂t
(∂
∂t
)= ∂
∂t
(∂
∂t∗dt∗
dt
)= ∂2
∂t∗2
(dt∗
dt
)2
= 6g
h0
∂2
∂t∗2, (A.1)
∂
∂x
(∂
∂x
)= ∂
∂x
(∂
∂x∗dx∗
dx
)= ∂2
∂x∗2
(dx∗
dx
)2
= 6
h20
∂2
∂x∗2 . (A.2)
because dt∗ = dt√6g/h0 and dx∗ = dx
√6/h0 from (4.1). This enables us to obtain
∂
∂tη =
(√6g
h0
∂
∂t∗
) (2h03
η∗)
= 2√6gh03
η∗t∗ , (A.3)
∂
∂xη =
(√6
h0
∂
∂x∗
) (2h03
η∗)
= 2√6
3η∗x∗ , (A.4)
η∂
∂xη =
(2h03
η∗) (
2√6
3η∗x∗
)
= 4h0√6
9η∗η∗
x∗ (A.5)
and
∂3
∂x2∂tη =
(6
h20
d2
dx∗2
) (√6g
h0
d
dt∗
) (2h03
η∗)
= 4
h0
√6g
h0η∗x∗x∗t∗ (A.6)
from (A.1) and (A.2). Finally, we substitute (A.3–A.6) to (1.1) to arrive at
2√6gh03
η∗t∗ + c0
(2√6
3η∗x∗
)
+ 3c02h0
(4h0
√6
9η∗η∗
x∗
)
− h206
(4
h0
√6g
h0η∗x∗x∗t∗
)
= 0,
(A.7)
where c0 = √gh0 from (1.2). It immediately follows that (A.7) is equivalent to (4.2).
Appendix B: Proof that the Iteration (3.14) is Essentially a Real-Valued(Recurrence) Equation if α is Real
Recall that (3.1) is equivalent to (2.26), which can be written as
η = I1 + I2 (B.1)
where
I1 = 1
2π
∫ ∞
−∞η1(k) · e−i θ (x,t;k)dk (B.2)
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and
I2 = 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· e−i θ (x−ξ,t−τ ;k) · ϕ (ξ, τ ) dξdkdτ . (B.3)
From (2.17),
I1 = 1
2π
∫ ∞
−∞
∫ ∞
−∞e−i[k(x−ξ)−ωBBM(k)·t] · η1 (ξ) dξdk
= 1
2π
∫ ∞
−∞
∫ ∞
−∞η1(ξ) · exp
[
−i
{
k (x − ξ) − c0k − α/ i
1 + h20k2/6
· t}]
dξdk
= 1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
· η1(ξ) · exp[
−i
{
k (x − ξ) − c0kt
1 + h20k2/6
}]
dξdk
= 1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
· η1(ξ) ·[
cos
[
k (x − ξ) − c0kt
1 + h20k2/6
]
−i sin
[
k (x − ξ) − c0kt
1 + h20k2/6
]]
dξdk
= 1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
· cos[
k (x − ξ) − c0kt
1 + h20k2/6
]
· η1(ξ)dξdk
− i1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
· sin[
k (x − ξ) − c0kt
1 + h20k2/6
]
· η1(ξ)dξdk.
(B.4)
where pseudo-dispersion relation (2.12) and pseudo-phase function (2.19) are used.Similarly as above, from (2.23),
I2 = 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· e−i θ (x−ξ,t−τ ;k) · ϕ (ξ, τ ) dξdkdτ
= 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−i
{
k (x − ξ) − c0k − α/ i
1 + h20k2/6
· (t − τ)
}]
·ϕ (ξ, τ ) dξdkdτ
= 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−α
1 + h20k2/6
· (t − τ)
]
· exp[
−i
{
k (x − ξ) − c0k
1 + h20k2/6
· (t − τ)
}]
· ϕ (ξ, τ ) dξdkdτ
= 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−α
1 + h20k2/6
· (t − τ)
]
· cos[
k (x − ξ) − c0k
1 + h20k2/6
· (t − τ)
]
· ϕ (ξ, τ ) dξdkdτ
123
1530 J Sci Comput (2018) 74:1504–1532
− i1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−α
1 + h20k2/6
· (t − τ)
]
· sin[
k (x − ξ) − c0k
1 + h20k2/6
· (t − τ)
]
· ϕ (ξ, τ ) dξdkdτ. (B.5)
Bear in mind that (3.2) can be written as
ψ = ∂I1∂x
+ ∂I2∂x
. (B.6)
From (B.4),
∂I1∂x
= 1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
· ∂
∂xcos
[
k (x − ξ) − c0kt
1 + h20k2/6
]
· η1(ξ)dξdk
−i1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
· ∂
∂xsin
[
k (x − ξ) − c0kt
1 + h20k2/6
]
· η1(ξ)dξdk
= − 1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
·k sin[
k (x − ξ) − c0kt
1 + h20k2/6
]
· η1(ξ)dξdk
−i1
2π
∫ ∞
−∞
∫ ∞
−∞exp
[−αt
1 + h20k2/6
]
·k cos[
k (x − ξ) − c0kt
1 + h20k2/6
]
· η1(ξ)dξdk. (B.7)
In the same way as above, differentiating (B.5) yields
∂I2∂x
= 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−α
1 + h20k2/6
· (t − τ)
]
· ∂
∂xcos
[
k (x − ξ) − c0k
1 + h20k2/6
· (t − τ)
]
·ϕ (ξ, τ ) dξdkdτ
−i1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−α
1 + h20k2/6
· (t − τ)
]
· ∂
∂xsin
[
k (x − ξ) − c0k
1 + h20k2/6
· (t − τ)
]
·ϕ (ξ, τ ) dξdkdτ
123
J Sci Comput (2018) 74:1504–1532 1531
= − 1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−α
1 + h20k2/6
· (t − τ)
]
·k sin[
k (x − ξ) − c0k
1 + h20k2/6
· (t − τ)
]
·ϕ (ξ, τ ) dξdkdτ
−i1
2π
∫ t
0
∫ ∞
−∞
∫ ∞
−∞1
1 + h20k2
6
· exp[
−α
1 + h20k2/6
· (t − τ)
]
·k cos[
k (x − ξ) − c0k
1 + h20k2/6
· (t − τ)
]
·ϕ (ξ, τ ) dξdkdτ.
(B.8)
The imaginary parts of (B.4–B.8) all vanish, if α is real, because the integrands of theimaginary parts of (B.4–B.8) are odd in k. Thus, the pair of (3.1) and (3.2) are all real-valued equations, which implies the iteration (3.14) is essentially a real-valued (recurrence)equation, since (3.11) is identical with the pair. This completes the proof.
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